Question

A random sample of 100 observations from a population with standard deviation 63 yielded a sample mean of 111. Complete parts a through c. a. Test the null hypothesis that y 100 against the alternative hypothesis that > 100, using a 0.05. Interpret the results of the test. Ho is rejected Ho is not rejected O Interpret the results of the test. Choose the correct interpretation below. O A. There is sufficient evidence to indicate the true population mean is not equal to 100 at O B. There is sufficient evidence to indicate the true population mean is greater than 100 at OC. There is sufficient evidence to indicate the true population mean is smaller than 100 at a 0.05. 0.05 0.05. b. Test the null hypothesis that u = 100 against the alternative hypothesis that u #100, using a=0.05. Interpret the results of the test. O O Ho is rejected. Ho is not rejected Interpret the results of the test. Choose the correct interpretation below. O A. There is insufficient evidence to indicate u is smaller than 100 at a 0.05. OB. There is sufficient evidence to indicate ju is greater than 100 at 0.05 O C. There is insufficient evidence to indicate fi is not equal to 100 at a = 0.05. c. Compare the results of the two tests you conducted. Explain why the results differ. Choose the correct answer below. O A. The results differ because the alternative hypothesis in part a is more specific than the one in b B. The results differ because the alternative hypothesis in part bis more specific than the one in a O Click to select your answer.

Answer 1

a)

The provided sample mean is 111 and the known population standard deviation is 63, and the sample size is n = 100

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ=100

Ha: μ>100

This corresponds to a right-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is z_c = 1.64

(3) Test Statistics

The z-statistic is computed as follows:

$z = \frac{\bar X - \mu_0}{\sigma/\sqrt{n}} = \frac{ 111 - 100}{ 63/\sqrt{ 100}} = 1.746$  

(4) Decision about the null hypothesis

Since it is observed that z = 1.746 > z_c = 1.64, it is then concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is greater than 100, at the 0.05 significance level.

A) Ho is rejected

B) There is sufficient evidence to claim that the population mean μ is greater than 100, at the 0.05 significance level.

b)

The provided sample mean is 111 and the known population standard deviation is 63, and the sample size is n = 100

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ=100

Ha: μ≠​100

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is z_c = 1.96

(3) Test Statistics

The z-statistic is computed as follows:

$z = \frac{\bar X - \mu_0}{\sigma/\sqrt{n}} = \frac{ 111 - 100}{ 63/\sqrt{ 100}} = 1.746$

(4) Decision about the null hypothesis

Since it is observed that |z| = 1.746 < z_c = 1.966, it is then concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ is different than 100, at the 0.05 significance level.

B) Ho is not rejected

C) There is insufficient evidence to claim that the population mean μ is different than 100, at the 0.05 significance level.

c)

Since, in a) the alternative hypothesis is only one sided where as in b) it is two sided

The correct answer is

B) The result differ because the alternative hypothesis in part b is more specific than the one in a)